Slope Fields and Euler’s Method Copyright © Cengage Learning. All rights reserved Day

2 Save The Date 2/25/2014  MOCK AP TEST ON TUESDAY 2/25  7:30-11:00  ATTENDANCE IS MANDATORY

3 Save The Date 3/25/2014  CALCULUS THE MUSICAL  Show begins at 1:00  We would leave at 12:20 and return back to school at 2:30.

4 6.1 Day 2: Slope Fields Greg Kelly, Hanford High School, Richland, Washington

5 A little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given: find We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant. This is the general solution

6 If we have some more information we can find C. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

7 Slope Fields

8 Solving a differential equation analytically can be difficult or even impossible. However, there is a graphical approach you can use to learn a lot about the solution of a differential equation. Consider a differential equation of the form y' = F(x, y) Differential equation where F(x, y) is some expression in x and y. At each point (x, y) in the xy–plane where F is defined, the differential equation determines the slope y' = F(x, y) of the solution at that point.

9 Slope Fields If you draw short line segments with slope F(x, y) at selected points (x, y) in the domain of F, then these line segments form a slope field, or a direction field, for the differential equation y' = F(x, y). Each line segment has the same slope as the solution curve through that point. A slope field shows the general shape of all the solutions and can be helpful in getting a visual perspective of the directions of the solutions of a differential equation. Slope fields are graphical representations of a differential equation which give us an idea of the shape of the solution curves. The solution curves seem to lurk in the slope field.

10 Slope Fields A slope field shows the general shape of all solutions of a differential equation.

11 Sketching a Slope Field Sketch a slope field for the differential equation by sketching short segments of the derivative at several points.

12 Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of

13 If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola. Slope fields show the general shape of all solutions of a differential equation. We can see that there are several different parabolas that we can sketch in the slope field with varying values of C.

14 Slope Fields  Create the slope field for the differential equation Since dy/dx gives us the slope at any point, we just need to input the coordinate: At (-2, 2), dy/dx = -2/2 = -1 At (-2, 1), dy/dx = -2/1 = -2 At (-2, 0), dy/dx = -2/0 = undefined And so on…. This gives us an outline of a hyperbola

15 Given: Let’s sketch the slope field … Slope Fields

16 Separate the variables Given f(0)=3, find the particular solution.

17 C Slope Fields In order to determine a slope field for a differential equation, we should consider the following: i) If points with the same slope are along horizontal lines, then DE depends only on y ii) Do you know a slope at a particular point? iii) If we have the same slope along vertical lines, then DE depends only on x iv) Is the slope field sinusoidal? v) What x and y values make the slope 0, 1, or undefined? vi) dy/dx = a( x ± y ) has similar slopes along a diagonal. vii) Can you solve the separable DE? 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ Match the correct DE with its graph: AB C E G D F H H B F D G E A

18 Slope Fields  Which of the following graphs could be the graph of the solution of the differential equation whose slope field is shown?

19  1998 AP Question: Determine the correct differential equation for the slope field: Slope Fields

20 Homework  Slope Fields Worksheet

21 Euler’s Method – BC Only Euler’s Method is a numerical approach to approximating the particular solution of the differential equation y' = F(x, y) that passes through the point (x 0, y 0 ). From the given information, you know that the graph of the solution passes through the point (x 0, y 0 ) and has a slope of F(x 0, y 0 ) at this point. This gives you a “starting point” for approximating the solution.

22 Euler’s Method From this starting point, you can proceed in the direction indicated by the slope. Using a small step h, move along the tangent line until you arrive at the point (x 1, y 1 ) where x 1 = x 0 + h and y 1 = y 0 + hF(x 0, y 0 ) as shown in Figure 6.6. Figure 6.6

23 Euler’s Method If you think of (x 1, y 1 ) as a new starting point, you can repeat the process to obtain a second point (x 2, y 2 ). The values of x i and y i are as follows.

24 Example 6 – Approximating a Solution Using Euler’s Method Use Euler’s Method to approximate the particular solution of the differential equation y' = x – y passing through the point (0, 1). Use a step of h = 0.1. Solution: Using h = 0.1, x 0 = 0, y 0 = 1, and F(x, y) = x – y, you have x 0 = 0, x 1 = 0.1, x 2 = 0.2, x 3 = 0.3,…, and y 1 = y 0 + hF(x 0, y 0 ) = 1 + (0 – 1)(0.1) = 0.9 y 2 = y 1 + hF(x 1, y 1 ) = (0.1 – 0.9)(0.1) = 0.82 y 3 = y 2 + hF(x 2, y 2 ) = (0.2 – 0.82)(0.1) =

25 Example 6 – Solution Figure 6.7 The first ten approximations are shown in the table. cont’d You can plot these values to see a graph of the approximate solution, as shown in Figure 6.7.

26 Homework  Slope Fields Worksheet  BC add pg odd